Background

In cloud physics there is an equation, usually called the growth
equation, which describes how a cloud droplet grows. Rogers and
Yau in their A Short Course in Cloud Physics provide a derivation
of this equation. Although overall Rogers and Yau's text is excellent,
in the matter of the derivation of the growth equation their presentation
is incomplete and points confusing because it misleads the reader. What
follows is an attempt to provide a more comprehensive derivation. The
presentation of the derivation differs from that of Rogers and Yau's in
notation. The generic radial distance variable is denoted by the lower
case symbol r and upper case R is used to denote the particular radius of
the droplet. This is the opposite of Rogers and Yau. The environmental
levels of water vapor density and temperature are identified by a
subscript e, in contrast to Rogers and Yau who denote the environmental
levels by the absence of a subscript. The values of water vapor
density and temperature are denoted here by a subscript R.

The Analysis

Cloud droplets may grow by condensation or diminish by evaporation.
Both condensation and evaporation are diffusion processes described
by the Fick diffusion equation; i.e., the flux of water vapor passing
through a plane M is given by:

M = -D∇ρ

where D is the diffusion coefficient for water vapor and ρ
is the density of water vapor in the space surrounding the droplet.
The negative sign in the above equation reflects the fact that
the mass flow is in the opposite direction from the gradient of water
vapor density.

When water vapor diffuses into the droplet it brings heat energy
which raises the temperature of the droplet and that heat energy may
diffuse out of the droplet into the surrounding moist air. Such
diffusion of heat energy obeys the same form of equation as does the
diffusion of the water vapor. The heat energy flux Q passing through
a plane is given by

Q = -K∇T.

When the net flow of water vapor or heat into or out of an infinitesimal
volume is considered they must generate a change in the density
and the temperature within that infinitesimal volume; i.e.,

∂ρ/∂t = D∇2ρ
and
c∂T/∂t = K∇2T

Under steady-state conditions (∂ρ/∂t = 0 and ∂T/∂t = 0)
the vapor density and temperature fields in the space surrounding the
droplet must satisfy the equations:

∇2ρ = 0
and
∇2T = 0.

In spherical coordinates under conditions of spherical symmetry
the Laplacian ∇2 is:

(1/r2)(∂(r2∂ /∂r)/∂r = 0

where r is the distance from the center of the droplet.

For a droplet of radius R the boundary conditions are:

ρ = ρR and T = TR at r=R
and
ρ -> ρe and T -> Te as r -> ∞

where ρe and Te are the environmental vapor density and temperature,
respectively.

The solution to the steady-state diffusion equation reduces to
integrating

∂(r2∂ρ/∂r)/∂r = 0

once with respect to r to get

r2∂ρ/∂r = C1

where C1 is a constant.

Dividing by r2 gives the equation

∂ρ/∂r = C1/r2

Integating this with respect to r gives

ρ = -C1/r + C2

The integration constants must be chosen to satisfy the
boundary conditions.

ρe = C2
and
ρR = -C1/R + C2

Subtracting the second equation from the first gives

(ρe-ρR) = C1/R
and thus
C1 = R(ρe-ρR)

The solution to the boundary value problem for the steady-state
diffusion equation is therefore

ρ(r) = ρe - (R/r)(ρe-ρR)

This equation reflects the fact that if ρe is greater
than ρR then ρ is decreasing smoothly as r decreases
toward R.

Temperature obeys the same equations as water vapor density so

T(r) = Te - (R/r)(Te-TR)
which can be rearranged to

T(r) = Te + (R/r)(TR-Te).

The latter form reflects the fact that if
TR is greater than Te then T increases smoothly
as r decreases to R.

The heat energy supplied to the droplet from the condensation of
water vapor into the droplet is:

LM = 4πLD(ρe-ρR)R

For equilibrium this must match the heat energy Q diffused out of the droplet; i.e.,

LM = Q
4πLDR(ρe-ρR) = 4πKR(TR-Te)
and hence
(TR-Te)/(ρR-ρe) = -LD/K

This above equation may be solved for the unknown ρR
in terms of the unknown TR as

ρR = ρe - (K/LD)(TR-Te)

The water vapor density at the surface of the droplet, ρR,
also obeys the ideal gas law so

ρR = e's(TR)/RVTR

where RV is the gas constant for water vapor and
e's(TR) is the saturated vapor pressure at
the temperature TR taking into account the Kelvin effect
due to the curvature of the droplet.

From the Kelvin Equation and the Raoult effect e's is given by

e's = es[1 + a/R - b/R3]

where es is the vapor pressure of water over an
infinite plane.

Combining the two equations for ρR gives

(es/RVTR)[1 + a/R - b/R3] =
(ρe - (K/LD)(TR-Te)

Dividing by (es/RVTR)
gives:

[1 + a/R - b/R3] = (ρe -
(K/LD)(TR-Te)RVTR/es(TR)

The RHS of the above equation is a function of the known parameters
and TR; the LHS is a cubic equation in (1/R). Thus for
given ρe and Te the solution for R for any
value of TR can be found. From the value of TR
the value of ρR can be found. It is, of course, the
inverse relationships of TR and ρR as functions
of R that are desired but that is a mere mathematical transformation.

B.J. Mason's analytical approximation of the solution to the
equation

Let ρVS and eS be the density and vapor pressure
of water vapor under saturated conditions. The Clasius-Clapyeron equation
is usually expressed in the form:

deS/dT = L/(T(αVS-αL))

where L is the latent heat of fusion and the α's refer to the
specific densities of water vapor and liquid water under conditions of
saturation. The specific volumes are
just the reciprocals of the densities. Therefore the Clasius-Clapyeron
equation can be expressed as:

deS/dT = LρVS/(T(1-ρVS/ρL))

Since ρVS in negligible compared to
ρL the above equation reduces to:

deS/dT = LρVS/T

But water vapor satisfies the ideal gas law

eS = ρVSRVT
and so
deS/dT = (dρVS/dT)RVT + ρVSRV

Equating the two expressions for deS/dT and dividing both sides
by ρVSRVT gives: